Gibbs Phenomenon of Framelet Expansions and Quasi-projection Approximation

TL;DR

This paper investigates the Gibbs phenomenon in framelet expansions and quasi-projection operators, establishing new identities and characterizations that show the phenomenon appears at all points for certain framelets and operators, improving prior results.

Contribution

It introduces a key identity for quasi-projection operators and characterizes the Gibbs phenomenon at arbitrary points, extending understanding to framelet expansions and general quasi-projection approximation.

Findings

Gibbs phenomenon occurs at all points for tight or dual framelets with at least two vanishing moments.

Gibbs phenomenon appears at all points for quasi-projection operators with at least three accuracy orders.

The results improve existing literature on wavelet expansions and are new for framelet and quasi-projection approximation.

Abstract

The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the $n$th Fourier partial sums overshoot a target function at jump discontinuities in such a way that such overshoots do not die out as $n$ goes to infinity. The Gibbs phenomenon for wavelet expansions using (bi)orthogonal wavelets has been studied in the literature. Framelets (also called wavelet frames) generalize (bi)orthogonal wavelets. Approximation by quasi-projection operators are intrinsically linked to approximation by truncated wavelet and framelet expansions. In this paper we shall establish a key identity for quasi-projection operators and then we use it to study the Gibbs phenomenon of framelet expansions and approximation by general quasi-projection operators. We shall also study and characterize the Gibbs phenomenon at an arbitrary point for approximation by quasi-projection operators. As a consequence, we show that the Gibbs phenomenon appears at all points for every tight or dual framelet having at least two vanishing moments and for quasi-projection operators having at least three accuracy orders. Our results not only improve current results in the literature on the Gibbs phenomenon for (bi)orthogonal wavelet expansions but also are new for framelet expansions and approximation by quasi-projection operators.